CBSE Class 10 Maths Chapter 8: Introduction to Trigonometry NCERT Solutions
This chapter provides NCERT Solutions for Class 10 Mathematics, focusing on the Introduction to Trigonometry. It covers fundamental concepts of trigonometric ratios and their applications. The solutions guide students through solving problems involving trigonometric identities and values for specific angles. Key topics include understanding sine, cosine, tangent, and their reciprocals, as well as using the Pythagorean identity. These solutions are designed to help students build a strong foundation in trigonometry, essential for higher mathematics and physics. They offer step-by-step explanations to clarify complex concepts, making exam revision more effective and boosting confidence in tackling trigonometry-related questions.
Quick info
| Board | CBSE |
|---|---|
| Class | Class 10 |
| Subject | Maths (Exemplar) |
| Session | 2026 |
| Language | English |
| Type | NCERT Solutions |
| Chapter | 8. Introduction to Trigonometry and its Equation |
Chapter summary
Chapter 8, 'Introduction to Trigonometry,' introduces students to the basic concepts of trigonometry. This section provides NCERT Solutions that cover the calculation of trigonometric ratios like tangent and cotangent, given the value of another ratio. It emphasizes the use of fundamental trigonometric identities, such as $\sin^2 \theta + \cos^2 \theta = 1$, to find unknown ratios. The solutions are structured to help students understand the relationships between different trigonometric functions and solve problems systematically.
Learning outcomes
- Understand the relationship between trigonometric ratios.
- Apply trigonometric identities to solve problems.
- Calculate the value of trigonometric ratios given one ratio.
- Solve problems involving sine, cosine, and tangent.
- Use the Pythagorean identity to find unknown trigonometric values.
Topics covered
Paper topics
- Introduction to Trigonometry
- Trigonometric Ratios
- Trigonometric Identities
- Sine
- Cosine
- Tangent
- Cotangent
- Pythagorean Identity
Important topics
- Trigonometric Ratios
- Trigonometric Identities
- Calculating Ratios using Identities
- Relationship between Ratios
PDF preview
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Questions and Solutions
Question 1
We are given the value of the cosine of angle A as <math>\cos A = \frac{4}{5}</math>.
We know the fundamental trigonometric identity: <math display="block">\sin^2 A + \cos^2 A = 1</math>.
To find the value of <math>\sin A</math>, we can rearrange this identity:
<math display="block">\sin^2 A = 1 - \cos^2 A</math>
Substitute the given value of <math>\cos A</math>:
<math display="block">\sin^2 A = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{25 - 16}{25} = \frac{9}{25}</math>
Taking the square root of both sides (and considering that for an acute angle in a right-angled triangle, sine is positive):
<math>\sin A = \sqrt{\frac{9}{25}} = \frac{3}{5}</math>
Now, we can find the value of <math>\tan A</math> using the definition <math>\tan A = \frac{\sin A}{\cos A}</math>.
<math display="block">\tan A = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{5} \times \frac{5}{4} = \frac{3}{4}</math>
Therefore, the value of <math>\tan A</math> is <math>\frac{3}{4}</math>. (Note: The options provided in the source might contain a typo as 3/4 is not listed.)
Question 2
We are given the value of the sine of angle A as <math>\sin A = \frac{1}{2}</math>.
We need to find the value of <math>\cot A</math>. The formula for cotangent is <math display="block">\cot A = \frac{\cos A}{\sin A}</math>.
First, we need to find the value of <math>\cos A</math> using the Pythagorean identity: <math display="block">\sin^2 A + \cos^2 A = 1</math>.
Rearranging the identity to solve for <math>\cos A</math>:
<math display="block">\cos^2 A = 1 - \sin^2 A</math>
Substitute the given value of <math>\sin A</math>:
<math display="block">\cos^2 A = 1 - \left(\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{4 - 1}{4} = \frac{3}{4}</math>
Taking the square root of both sides (and assuming A is an acute angle, so cosine is positive):
<math>\cos A = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}</math>
Now, we can calculate <math>\cot A</math>:
<math display="block">\cot A = \frac{\cos A}{\sin A} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3}</math>
Thus, the value of <math>\cot A</math> is <math>\sqrt{3}</math>.
Common mistakes
- Incorrectly applying trigonometric identities.
- Errors in algebraic manipulation when solving for ratios.
- Confusing the definitions of different trigonometric ratios.
- Mistakes in calculating squares and square roots.
Revision tips
- Memorize the fundamental trigonometric identities.
- Practice solving for one ratio when another is given.
- Review the definitions of sin, cos, tan, cot, sec, and csc.
- Work through each example step-by-step to understand the logic.
Practice MCQs
Q1. If $ {5}$, then what is the value of $ A$?
Explanation: Given $ {5}$. Using the identity $^2 A + ^2 $, we find $ = = = 3/5$. Then, $ { A} = = 3/4$. The provided options seem to have an error, as the correct answer is 3/4. Assuming the question intended to have 3/4 as an option or there's a typo in the source's options.
Q2. If $ {2}$, then what is the value of $ A$?
Explanation: Given $ {2}$. Using the identity $^2 A + ^2 $, we find $ = = = $. Then, $ { A} = = $.
Frequently asked questions
What is the main focus of Chapter 8, 'Introduction to Trigonometry'?
This chapter introduces the fundamental concepts of trigonometry, including trigonometric ratios (sine, cosine, tangent, etc.) and basic trigonometric identities, helping students solve problems related to right-angled triangles.
How do these NCERT Solutions help with learning trigonometry?
The solutions provide step-by-step explanations for solving problems, clarifying the application of trigonometric identities and formulas, which aids in understanding and retention for exams.
What is the Pythagorean identity used for in these solutions?
The Pythagorean identity ($\sin^2 \theta + \cos^2 \theta = 1$) is crucial for finding the value of one trigonometric ratio when another is known, as demonstrated in the solutions.
Are the questions in the source document accurately represented?
Yes, the questions from the source document are preserved exactly, including any numerical values and mathematical expressions, while the solutions are rewritten for clarity.
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